Associated to a fine and saturated log stack \(X\) , we have the tautological morphism \(X \rightarrow \textbf {Log}\)  which factors through a unique log algebraic stack \(X \rightarrow \mathcal {X}\) , with \(\mathcal {X} \rightarrow \textbf {Log}\)  representable and étale. We call the log stack \(\mathcal {X}\)  the Artin fan of \(X\) . See [1]} section 3 for further details. Specializing to the case in which \(X\)  is a log smooth scheme, the moduli problem over schemes of basic families of punctured log maps to \(\mathcal {X}\)  defines an algebraic log stack \(\mathfrak {M}(\mathcal {X},\tau )\)  of prestable punctured curves equipped with a marking by \(\tau \) . This log stack is equipped with an idealized log structure described in [2]} Definition \(3.22\) . We have a morphism \(\mathcal {M}(X,\tau ) \rightarrow \mathfrak {M}(\mathcal {X},\tau )\)  given by composing a punctured log curve \(C \rightarrow X\)  with \(X \rightarrow \mathcal {X}\) . Moreover, unlike \(\mathcal {M}(X,\tau )\)  in general, \(\mathfrak {M}(X,\tau )\)  is idealized log smooth by [2]} Theorem \(3.24\) . Furthermore, there is an obstruction theory associated to the morphism \(\mathcal {M}(X,\tau ) \rightarrow \mathfrak {M}(\mathcal {X},\tau )\) , which facilitates virtual pullback in the sense of [4]}.
